As for the other links \(L\)  encountered so far, there exists an invariance of the card seq of \(\pi _1(L)\)  when \(L=\tilde{D}_4\) . Let us apply the map \(a \rightarrow b\) , \(b \rightarrow abc\) , \(c \rightarrow a\)  on the 3 generators of \(L\) , the substitution map \(T = \begin{pmatrix} 0 &1 &1\\ 1 &1 & 0 \\ 0 &1 & 0\end{pmatrix}\)  is primitive since \(T^3>>0\)  and the Perron-Frobenius eigenvalue is the real root of the polynomial \(\lambda ^3-2 \lambda +1=0\) , that is \(\lambda _{PF} \sim 1.83928\) , the Tribonacci constant [1]}, see also [2]} and [3]} for mathematical details. This reveals the aperiodicity of the fundamental group.
